Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{2n^2 - 24n + 40}{2n^3 + 2n^2 - 12n}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {2(n^2 - 12n + 20)} {2n(n^2 + n - 6)} $ $ x = \dfrac{2}{2n} \cdot \dfrac{n^2 - 12n + 20}{n^2 + n - 6} $ Simplify: $ x = \dfrac{1}{n} \cdot \dfrac{n^2 - 12n + 20}{n^2 + n - 6}$ Next factor the numerator and denominator. $ x = \dfrac{1}{n} \cdot \dfrac{(n - 2)(n - 10)}{(n - 2)(n + 3)}$ Assuming $n \neq 2$ , we can cancel the $n - 2$ $ x = \dfrac{1}{n} \cdot \dfrac{n - 10}{n + 3}$ Therefore: $ x = \dfrac{ n - 10 }{ n(n + 3)}$, $n \neq 2$